Linear multiplicative functionals of algebras of $S$-analytic functions on groups

Let $S$ be a subsemigroup of a semigroup $\sum$ that generates a group $G$. We find conditions that assure extendability of linear multiplicative functionals of the algebra $A_S$ of $S$-analytic functions on $G$ with spectra in $S$ to linear multiplicative functionals of the corresponding algebra $A_{\sum}$. Conditions for existence of dense homeomorphic embeddings of the upper half plane in the maximal ideal space of the algebra of almost periodic functions on $\mathbb R$ with spectrum in $S$, and of the open unit disc in the maximal ideal space of certain subalgebras of $H^\infty$ are obtained as corollaries.